👥 We’re introducing Samuel Orso and Aleksandr Shemendyuk with profiles detailing roles, hobbies, quotes, schedule, and pictures. It’s a sneak peek into their life!
📊 Want to learn R Markdown syntax? We offer practical examples, discussing caching, sample creation, result comparison, and making histograms.
🧮 Finding equations difficult? We’ll guide you on embedding equations seamlessly into your text, with plenty of examples.
🎨 We’ll also explore fun text formatting techniques like creating buttons and colorful boxes.
📚 For a complete guide, check the final reference section – a must-have companion for this journey.
Lecturer in Data Science at UNIL,…
My hobbies are…💡
“Some mathematician, I believe, has said that true pleasure lies not in the discovery of truth, but in the search for it.” – Tolstoi
| Class | Time |
|---|---|
| PTDS | 8h30 - 12h00 |
Hello everyone! I’m Aleksandr, a PhD student in Actuarial Science and your teaching assistant. When I’m not diving into research, you might find me snowboarding down slopes 🏂, trekking up mountains 🏔, tinkering with AI 🤖, or playing chess ♟. I’m dedicated and always aim for excellence. Let’s make this academic journey memorable together! 🌟📚
“As far as you can avoid it, do not give grief to anyone. Never inflict your rage on another. If you hope for eternal rest, feel the pain yourself; but don’t hurt others.” – Omar Khayyám
| Class | Time |
|---|---|
| PTDS | 8h30 - 12h00 |
Example of a situation where the cache = T leads to
incorrect calculations:
## [1] -0.5604756
This chunk will store a randomly generated number. However, because
we set cache = T, when we rerun the chunk, the number won’t
change as it will be retrieving the number from the cache.
## [1] -0.5604756
In the chunk above, if we need to get a different number, we will get
the same number every time we run it because we used
cache = T in the first chunk. This might not be beneficial
especially in situations where we require different outputs every time
we run the code.
set.seed(123)
# Generate 2000 random samples
lambda <- 2
y <- rpois(2000, lambda)
# Compute empirical mean and variance
y_mean <- mean(y)
y_var <- var(y)
# Theoretical mean and variance of a Poisson distribution
theoretical_mean <- 2
theoretical_variance <- 2
# Compare
print(paste("Empirical mean: ", y_mean))## [1] "Empirical mean: 1.9915"
## [1] "Theoretical mean: 2"
## [1] "Empirical variance: 1.97441495747874"
## [1] "Theoretical variance: 2"
The empirical mean and variance should be close to the theoretical value (\(\lambda = 2\)) but might not match precisely. This is because of the randomness inherent in the simulation. With larger sample sizes, the empirical values should get closer to the theoretical values due to the Law of Large Numbers.
| Measure | Sample estimate | Theoretical | Absolute difference |
|---|---|---|---|
| Mean | 1.992 | 2 | 0.008 |
| Variance | 1.974 | 2 | 0.026 |
To display the histogram of \(y\), one would use the following command:
Historgram of y with 10 bins
adding as an option to this chunk the option
fig.cap="Historgram of y with 10 bins" to display the
caption to this figure. Here, we do not observe the accurate shape of
the Poisson distribution with \(\lambda =
2\), which should be more centred around the value of 2.
Historgram of y with 50 bins
Increasing the number of bins to 50 results in a more granular view of the data distribution. However, it could also lead to overfitting where too much noise is mistaken for an actual pattern. It’s important to choose an appropriate number of bins to accurately convey the distribution of the data in a histogram.
Include the following equation
\[ f(x;\, \mu, \sigma^2) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(x-\mu)^2 / (2\sigma ^2)} \]
This formula represents the Probability Density Function of a Normal Distribution where \(\mu\) is the mean and \(\sigma^2\) is the variance.
Include the following in-line equation:
$ \hat{p}_{stress} =k s^{-m}$
This is a typical Stress-Life equation in material science, where \(\hat{p}_{stress}\) represents the predicted stress, \(s\) is the life of the material, and \(k\) and \(m\) are material constants.
On how to include references in Markdown, see the guide.
As shown in Choi, Marstaller, and Aceves (2023), the singular integral representation of the fractional Laplacian is defined as
\[ (-\Delta)^{\frac{\alpha}{2}} u(x)=c(d, \alpha) \text { p.v. } \int_{\mathbb{R}^d} \frac{u(x)-u(y)}{|x-y|^{d+\alpha}} d y:=\frac{2^\alpha \Gamma\left(\frac{\alpha}{2}+\frac{d}{2}\right)}{\pi^{\frac{d}{2}}\left|\Gamma\left(-\frac{\alpha}{2}\right)\right|} \text { p.v. } \int_{\mathbb{R}^d} \frac{u(x)-u(y)}{|x-y|^{d+\alpha}} d y . \]
\[ (-\Delta)^{\frac{\alpha}{2}} u(x)=c(d, \alpha) \text { p.v. } \int_{\mathbb{R}^d} \frac{u(x)-u(y)}{|x-y|^{d+\alpha}} d y:=\frac{2^\alpha \Gamma\left(\frac{\alpha}{2}+\frac{d}{2}\right)}{\pi^{\frac{d}{2}}\left|\Gamma\left(-\frac{\alpha}{2}\right)\right|} \text { p.v. } \int_{\mathbb{R}^d} \frac{u(x)-u(y)}{|x-y|^{d+\alpha}} d y . \]
<span style="color:blue">"The only thing that is constant is change.", Heraclitus.</span>
“The only thing that is constant is change.”, Heraclitus.
<div style="background-color: #F5F5DC; padding: 10px; border: 1px solid black;"> "Learning never exhausts the mind." - Leonardo da Vinci </div>
“Learning never exhausts the mind.” - Leonardo da Vinci